A Classification of \({\varvec{n}}\)-Tuples of Commuting Shifts of Finite Multiplicity

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Abstract

Let \(\mathbb {V}\) denote an n-tuple of shifts of finite multiplicity, and denote by \({{\mathrm{Ann}}}(\mathbb {V})\) the ideal consisting of polynomials p in n complex variables such that \(p(\mathbb {V})=0\). If \(\mathbb {W}\) on \(\mathfrak {K}\) is another n-tuple of shifts of finite multiplicity, and there is a \(\mathbb {W}\)-invariant subspace \(\mathfrak {K}'\) of finite codimension in \(\mathfrak {K}\) so that \(\mathbb {W}|\mathfrak {K}'\) is similar to \(\mathbb {V}\), then we write \(\mathbb {V}\lesssim \mathbb {W}\). If \(\mathbb {W}\lesssim \mathbb {V}\) as well, then we write \(\mathbb {W}\approx \mathbb {V}\). In the case that \({{\mathrm{Ann}}}(\mathbb {V})\) is a prime ideal we show that the equivalence class of \(\mathbb {V}\) is determined by \({{\mathrm{Ann}}}(\mathbb {V})\) and a positive integer k. More generally, the equivalence class of \(\mathbb {V}\) is determined by \({{\mathrm{Ann}}}(\mathbb {V})\) and an m-tuple of positive integers, where m is the number of irreducible components of the zero set of \({{\mathrm{Ann}}}(\mathbb {V})\).

Keywords

Constrained operators Commuting pure isometries Virtual similarity 

Mathematics Subject Classification

47A13 

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of ManitobaWinnipegCanada

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